Circular Current Loop as a Magnetic Dipole

Why a Current Loop is a Magnetic Dipole

A circular current loop behaves like a magnetic dipole when viewed from far away. This means its magnetic field pattern at large distances is identical to the electric field pattern of an electric dipole. The key idea is that the loop has a magnetic dipole moment $\mathbf{m}$, which is the magnetic analogue of the electric dipole moment $\mathbf{p}$.

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Derivation of the Magnetic Field at Large Distances

We start from the known result for the magnetic field on the axis of a circular loop of radius $R$, carrying current $I$, at a distance $x$ from its centre (from Eq. 4.11):

Step 1: Far-field approximation

For $x \gg R$, we can neglect $R^2$ in the denominator:

Step 2: Introduce area and magnetic moment

The area of the loop is $A = \pi R^2$. The magnetic dipole moment is defined as:

Substituting $R^2 = \frac{m}{\pi I}$ into the field expression:

Rewriting in a standard form:

This is the axial field of a magnetic dipole.

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Analogy with Electric Dipole

The electric field on the axis of an electric dipole (from Chapter 1) is:

Comparing the two expressions, we see a perfect analogy if we make the replacements:

• $p \rightarrow m$ (dipole moment)
• $\frac{1}{4\pi \varepsilon_0} \rightarrow \frac{\mu_0}{4\pi}$ (permittivity → permeability)

Thus, a current loop is the magnetic analogue of an electric dipole.

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Field in the Plane of the Loop (Equatorial Field)

For a point in the plane of the loop (perpendicular bisector of the dipole), at large distance $x \gg R$, the magnetic field is:

This matches the electric field on the perpendicular bisector of an electric dipole:

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Key Result: Any Planar Loop is a Magnetic Dipole

For any planar current loop (not just circular), the magnetic dipole moment is:

where $\mathbf{A}$ is the area vector (magnitude = area, direction = normal to the plane by right-hand rule). At large distances, the loop produces the same field as a point magnetic dipole with moment $\mathbf{m}$. …